Often, individuals will seek to guarantee a steady stream of income for situations when their regular income is either diminished or altogether unavailable (e.g. retirement, times of financial hardship, etc). Additionally, individuals may make an investment in order to expand their income at a later date. In such cases, these individuals will often invest their existing funds, in order to guarantee a stream of payments of the invested amount, as well as to receive return from the investment at a later time. This type of investment is generally known as an “annuity.” An individual investing in an annuity (and upon whose life the income payments will be based) is known as an “annuitant.” The individual receiving payments from the annuity is known as the “contract owner.” The contract owner and the annuitant are often the same individual.
The annuity has two phases: an accumulation phase, during which the contract builds a cash value and money is added, and a payout phase, during which the funds are distributed. An annuitant may choose to purchase an annuity with a lump sum, or may make continuous payments into an annuity fund. Regardless of the payment method chosen by the annuitant, the financial or insurance institution offering an annuity will begin making periodic payments to and stop receiving funds from the annuitant on a predetermined date, this is known as “annuitization.”
Two common types of annuities are known as fixed annuities (“FA”) and variable annuities (“VA”). Upon annuitization, fixed annuities offer payments of predetermined value, or of sums that increase by a set percentage. Conversely, upon annuitization, variable annuities offer payments determined by the performance of a particular investment option (e.g. bonds).
With a variable annuity, the annuity contract owner bears the investment risk. The relevant life typically has a choice of investment options in which he/she can direct where the annuity deposits will be invested. These various investment options, or sub-accounts, may include stocks, bonds, money market instruments, mutual funds and the like.
Since the yield of a variable annuity is dependent on the specific sub-accounts, the risk involved in purchasing a variable annuity is proportional to the risk involved in investing in the underlying sub-accounts. While a potential annuitant may be interested in a specific investment option, the risk involved over a long period of time before annuitization may be unappealing. In such a case, the financial institution or insurance company offering the variable annuity product may elect to guarantee a certain minimum return on the annuitant's investment. Thus, the financial or insurance institution would assume some of the risk involved in purchasing a variable annuity product.
Variable annuity contracts can also provide a death benefit. Usually, during the accumulation period, this death benefit is related to the value of the underlying sub-accounts, or contract value. That is, if the sub-accounts backing the contract value have performed poorly, then the death benefit may be reduced to an insignificant amount. After annuitization, the death benefit can be a function of the remaining payments of the annuity at the time of the relevant life's death.
Annuity contracts may also provide guarantees in several different variations. A guaranteed minimum death benefit (“GMDB”) is a guarantee that provides a minimum benefit at the death of the relevant life regardless of the performance of the underlying investments. A guaranteed minimum income benefit (“GMIB”) is a guarantee that will provide a specified minimum income amount at the time the contract is annuitized. The income payment will be dependent on previously stated details set out in the contract.
A guaranteed minimum accumulation benefit (“GMAB”) is a benefit that guarantees a specified minimum contract value at a certain date in the future, even if actual investment performance of the contract is less than the guaranteed amount. A guaranteed minimum withdrawal benefit (“GMWB”) is a guarantee of income for a specified period of time, and in some versions, the income stream is guaranteed for life without requiring annuitization as in the guaranteed minimum income benefit. However, this guarantee will automatically annuitize the contract if the contract value is reduced to zero or some other amount specified in the contract or rider (as described below). Further, if the annuity contract does not provide a guarantee (e.g. GMIB, GMWB, etc.), the contract will terminate when the contract value goes to $0 or some other amount specified in the contract.
The guarantees form a class of liabilities collectively referred to as variable annuity guaranteed benefits (“VAGB”). These guarantees are known as “riders.” The contract holder can purchase riders in order to guarantee a certain minimum performance criteria for underlying separate accounts. To the extent that the underlying investments do not perform in such a way so that the minimum criteria are met, the writing insurance company must subsidize the difference between the minimum performance that the VAGB guarantees and the actual performance of the underlying sub-accounts.
VAGBs can be further subdivided into several different categories, which are shown in FIG. 1. Two common categories of VAGBs 104 are variable annuity guaranteed minimum death benefits (“VAGMDB”) 106 and variable annuity guaranteed living benefits (“VAGLB”) 108.
The VAGMDB 106 is commonly implemented as one the following three (3) manifestations: return of premium death benefits (“ROPDB”) 110, high watermark and/or periodic ratchet death benefits (“HWDB/PRDB”) 112, and earnings enhancement death benefits (“EEDB”).
The VAGLB 108 is commonly implemented as one the following three (3) manifestations: variable annuity guaranteed minimum accumulation benefits (“GMAB”) 116, variable annuity guaranteed minimum income benefits (“GMIB”) 118, and variable annuity guaranteed minimum withdrawal benefits (“GMWB”) 120. This is not intended as an exhaustive list, but rather, a broad overview of the general trend of VAGBs currently available.
As previously discussed, two types of VAGBs are VAGMDB and VAGLB. The fundamental difference between VAGMDBs and VAGLBs is that the former requires that the annuitant die in order for the contract holder to realize the incremental value afforded by the guarantee. On the contrary, VAGLBs permit the contract holder to realize some or all of the benefit of the guarantee while the annuitant is living.
Naturally, regardless of the specific form taken by a particular VAGB, there is significant risk that the underlying sub-accounts perform in a manner that is inadequate to meet the minimum performance criteria. Financial and insurance institutions offering VAs are often interested in decreasing the risks entailed in the sale of a variable annuity.
Historically, many financial and insurance institutions have purchased reinsurance in an attempt to share the risk that there will be inadequate funds available to cover these guarantees. Typically, reinsurance companies spread risks by pooling the risks of multiple companies and contracts. Specifically, insurance companies pay a premium to cede a portion of their risk so that if losses are above a negotiated amount, the reinsurance company will reimburse the insurance company for these excess losses. Since the reinsurance company assumes risks from multiple companies, any losses incurred from business assumed from one insurance company are expected to be outweighed by profits from another company, thus allowing the reinsurance company to make a profit. Over the years, most reinsurers have withdrawn from providing coverage for variable annuity contracts having features such as those described above because of the high correlation among the contracts, and thus the risk could not be mitigated by pooling risks from multiple companies. Because of the inability of insurance companies to reinsure variable annuity contracts, they incurred large economic losses during the stock market decline from 2000 to 2002.
Another method of mitigating these risks is known as a “hedge,” while investing in a hedge is known as “hedging.” To protect the variable annuity contract holders, and to ensure the claims-paying ability of the writing insurance company, hedging programs are often maintained by insurance carriers to offset the risk associated with the riders.
Hedging is a strategy that entails making an investment, the gains of which will offset the losses of a business risk, thus allowing the hedging entity to benefit from a gain involved in a particular business transaction while offsetting losses. Commonly, hedging is used to diminish the risk factors involved in a specific investment, but can also be used to manage the risks involved in guaranteeing a minimum income to an annuitant on a variable annuity, or for a group of annuitants whose variable annuities depend on the same or similar factors.
Various strategies are being employed to manage GMWB and GMDB risks: (a) delta hedging (uncertain effectiveness since a GMWB's exposure to “vega” (implied volatility) is not hedgeable with futures, and substantial EPS (earnings per share) and economic risks remain); (b) Multi-greek hedging with futures and vanilla options (reasonably effective to hedge economic and EPS exposure over intermediate term and stable markets, but long term effectiveness is uncertain due to cost and exposure to second and third order risks); and (c) Multi-greek hedging with futures, vanilla options, and exotic equity options (reasonably effective to hedge EPS exposure over intermediate term and stable markets, but long term effectiveness is not well understood).
Hedging programs can vary significantly but generally proceed according to the following pseudo-algorithm:                Construction of a mathematical valuation model to compute an estimate of the value of the written guarantee liability, conditional upon a set of relevant capital markets data and assumptions for annuitant behavior.        Gathering required capital markets data, dependent upon the structure of the guarantee and the contractually permissible set of investment options, but generally including:                    The spot price of relevant equity indices.            The term structure of interest rates denominated in all of the currencies that are reflected in the valuation model.            The spot price of relevant cross-currency exchange rates associated with all of the currency pairs that are reflected in the valuation model.            The forward implied dividend curves for each of the relevant equity indices.            A sub-model for the volatility associated with the price of the equity indices.            A sub-model for the volatility associated with the relevant cross-currency exchange rates.            A sub-model for the volatility associated with the interest rates of all relevant term structure.                        Formulating assumptions of annuitant behavior, dependent upon the structure of the guarantee, but generally including:                    Assumed rates of mortality for individual annuitants, or a sub-model for stochastic mortality.            Assumed rates of lapsation for individual annuitants and/or a predefined algorithm (deterministic or stochastic) for future lapsation rates that is a function of other variables in the valuation (so-called “dynamic lapsation”).            Assumed rates of utilization for behavioral choices granted to the contract holder under the terms of the guarantee, such as size and frequency of periodic withdrawal of funds from the variable annuity contract and/or a predefined algorithm (deterministic or stochastic) for future utilization that is a function of other variables in the valuation (“dynamic utilization”).            Size, style and frequency of transfer of funds between investment options and/or a predefined algorithm (deterministic or stochastic) for future transfers that is a function of other variables in the valuation.                        Defining a series of sets of unexpected fluctuations (“shocks”) to be applied to capital markets data.        Running the valuation model and computing an estimate of the valuation of the written liability under the “base case” market data and under each set of shocks. This information can be used to determine an estimate of the base valuation of the written liability and of the sensitivities of the valuation estimate to changes in specific capital markets data. (The estimate of the base valuation of the written liability and the estimate of the sensitivities of the valuation estimate to changes in specific capital markets data are known in the art as “Greeks”).        Formulating a hedge portfolio and executing an analogous valuation/sensitivity exercise to calculate the base valuation and the Greeks.        Executing trades in the hedge portfolio that position the aggregated Greeks of the hedge portfolio to be within desired ranges relative to the liability Greeks.        Several useful notations are defined as follows:        ƒLB is the liability valuation under the base case set of capital markets assumptions        δL is the liability delta        κL is the liability vega        ρL is the liability rho        
Furthermore, it should be noted that the liability delta is defined as the sensitivity of the VAGB valuation to instantaneous changes in stock or stock index levels, the liability vega is defined as the sensitivity of the VAGB valuation to instantaneous changes in stock or stock index implied volatility levels, and the liability rho is defined as the sensitivity of the VAGB valuation to instantaneous changes in interest rates.
Additionally, there are innumerable other Greeks that can be defined and managed as part of a hedge program. Those may include the following additional metrics:                deltas with respect to foreign currency exchange rates,        “partial” or “bucket” vegas which are sensitivities of, the VAGB valuation to implied volatilities of a specific tenor rather than to the implied volatility surface as a whole,        “partial” or “key rate” rhos which are sensitivities of the VAGB valuation to interest rates of a specific tenor rather than to the entire yield curve in parallel,        “correlation vegas” which are the sensitivities of the VAGB valuation to changes in the level of correlation between set of capital markets variables assumed to be stochastic,        “theta,” which is the sensitivity of the VAGB valuation to the passage of time,        any number of higher-order sensitivities, and        any number of cross-gammas.        
Finally, it should be noted that the above is by no means meant to be an exhaustive list of all possible Greeks, but merely an illustrative description of some Greeks that may be instrumental in understanding the subject matter of the invention.
The following is an example of hedging the liability associated with a VA. Assuming a VA guarantee has been written on a VA contract in which the policyholder's funds are invested in a single asset, and supposing the writing insurance company wishes to implement a 3-Greek first order hedge to offset delta, vega, and rho risk. Let ƒL(x,y,z) be the liability valuation estimate computed after a hypothetical change to the underlying stock index of x, a hypothetical change to the underlying index volatility of y, and a hypothetical change to the underlying interest rate term structure of z.
In classic terminology, the three Greeks of the liability would be computed using a finite-differencing methodology as follows:ƒLB=ƒ(0,0,0)δL=ƒL(1,0,0)−ƒL(0,0,0)=ƒL(1,0,0)−ƒLB κL=ƒL(0,0.01,0)−ƒL(0,0,0)=ƒL(0,0.01,0)−ƒLB ρL=ƒL(0,0,0.0001)−ƒL(0,0,0)−ƒL(0,0,0.0001)−ƒLB 
Similar metrics can be calculated for a portfolio of hedging assets. Using analogous notation, a standard goal of a hedging program would be to formulate a portfolio of hedging assets that, minimally, meets the following criteria:
                                    δ          A                -                  δ          L                            <                  ɛ        δ            ⁢                          ⁢      and      ⁢              /            ⁢      or      ⁢                          ⁢              ɛ                  δ          ,          1                      <                  δ        A                    δ        L              <          ɛ              δ        ,        2                                                κ          A                -                  κ          L                            <                  ɛ        κ            ⁢                          ⁢      and      ⁢              /            ⁢      or      ⁢                          ⁢              ɛ                  κ          ,          1                      <                  κ        A                    κ        L              <          ɛ              κ        ,        2                                                ρ          A                -                  ρ          L                            <                  ɛ        ρ            ⁢                          ⁢      and      ⁢              /            ⁢      or      ⁢                          ⁢              ɛ                  ρ          ,          1                      <                  ρ        A                    ρ        L              <          ɛ              ρ        ,        2            
Wherein, the ε (“epsilon”) represents tolerance imposed on the management of the portfolio. Notably, they need not be constant, although they are expressed that way above. The above examples demonstrate first-order hedging only in a 3-Greek framework, but similar exercises may be performed related to higher-order Greeks and cross-gammas. That is, second-order Greeks and cross-gammas can be calculated using analogous finite differencing methodologies and analogous hedge tolerances can be defined. Furthermore, each of the above three equations represents the hedging of a different Greek (i.e. delta, vega, and rho). Additionally, the main difference between the first and second condition, in each of the above three equations, is whether the deviation in liability and asset Greeks is viewed on an absolute and/or a relative basis.
The motivation for Greek-matching is to produce gains or losses that offset losses or gains made on the liability. For example, suppose that the price of the underlying asset were to decrease by one. This would result in a change (generally a loss in the case of VA guarantees) of δL. However, due to the construction of the asset portfolio, the hedges will experience a change in value (generally a gain in the case of VA guarantee hedges) of δA. Since the hedging process ensures that the difference between δA and δL is small, the net economic impact on the company is also small, making the company reasonably indifferent to small changes in these capital market variables.
Among other factors, hedging effectiveness on VA guarantees using generic hedging instruments, known as “vanilla hedging instruments,” is dependent upon the size, frequency and correlation of movements in critical capital markets variables. Generally, small changes in valuation inputs will not cause a hedge to materially lose effectiveness. Depending upon the nature of the guarantee written, as well as upon the exact instruments chosen for hedging, there are two characteristics of VA guarantee liabilities that cause hedging ineffectiveness in existing systems for guaranteeing benefits using hedging: 1) the instance where valuation inputs experience large and/or sudden changes, and 2) when several of the inputs move together.